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15 June 2018

Low-frequency Arm and Cartridge Interaction, Part II

In our previous post, we've traced down the derivation of an effective mass/compliance/resonance formula to a 1954 AES article, and highlighted some differences between 1950s cartridges and later stereo ones.

Today, we'll examine these differences and will try to estimate their effect on the low frequency resonance.

Today, we'll examine these differences and will try to estimate their effect on the low frequency resonance.

And our first and most obvious difference, **flexure suspension in 1950s mono cartridges versus an elastomer "donut" in stereo cartridges.** Is elastomer that different from a spring?

Let's briefly recall a basic formula for harmonic oscillator's frequency,

$$\omega= \sqrt{k\over m}$$

(**k** is a spring constant and *m* is effective mass)

It's based on Hooke's law, which states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance. The charts showing the relationship between such force and a distance are called "Stress/Strain" charts.

We've got one of them on the right, and it shows typical stress (force) / strain (deflection or distance) relationships in a steel spring and a rubber band. In this particular case, the rubber is being stretched. While the spring isn't perfectly linear, the rubber isn't linear at all!

There are 3 distinct regions in rubber's reaction to a stretching force. First, some significant force is required to start the stretching. After a certain level, further stretching goes on without much additional force. And then we hit a soft limit of sorts, when applying additional force doesn't result in much of an extra movement.

$$\omega= \sqrt{k\over m}$$

(

It's based on Hooke's law, which states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance. The charts showing the relationship between such force and a distance are called "Stress/Strain" charts.

We've got one of them on the right, and it shows typical stress (force) / strain (deflection or distance) relationships in a steel spring and a rubber band. In this particular case, the rubber is being stretched. While the spring isn't perfectly linear, the rubber isn't linear at all!

There are 3 distinct regions in rubber's reaction to a stretching force. First, some significant force is required to start the stretching. After a certain level, further stretching goes on without much additional force. And then we hit a soft limit of sorts, when applying additional force doesn't result in much of an extra movement.

Assuming for now that rubber in our cartridge suspension is only being stretched, what would it mean for compliance?

First of all, it means that*our little formula no longer applies*. It's only derived for materials with fixed spring constant *k*, and with elastomers like rubber *k* isn't a constant any more. It varies a lot with applied force.

Second, it means that a cartridge with stretching elastomer suspension will have not one compliance, but three distinct ones. Relatively low one for small deflections, high compliance for medium deflections, and then again low compliance for large deflections at the limits of cantilever travel.

First of all, it means that

Second, it means that a cartridge with stretching elastomer suspension will have not one compliance, but three distinct ones. Relatively low one for small deflections, high compliance for medium deflections, and then again low compliance for large deflections at the limits of cantilever travel.

But the elastomer "donut" of cartridge's suspension doesn't only stretch under load. It's both stretched and compressed. How does the elastomer behave when it is compressed?

Here's the stress/strain chart for compression. It's a bit different, the area of "initial resistance" is usually absent. Elastomer first yields quite willingly, and then the resistance builds up.

If we add the stretch and compression stress/strain charts together, we'll get a picture of low-ish initial compliance, a relatively large region of high compliance, and then very low and rapidly falling compliance at the limit (red line on the illustration).

This fits well with observed behaviour of most modern stereo cartridges, at least in the horizontal plane.

This fits well with observed behaviour of most modern stereo cartridges, at least in the horizontal plane.

But what about the vertical plane, why should it be any different?

It is different because, unlike the horizontal plane, the "donut" is preloaded by downforce in the vertical plane. It's up to a cartridge designer to decide which compliance zone to preload it into. Most cartridges's vertical compliance is a lot lower than horizontal.

It is different because, unlike the horizontal plane, the "donut" is preloaded by downforce in the vertical plane. It's up to a cartridge designer to decide which compliance zone to preload it into. Most cartridges's vertical compliance is a lot lower than horizontal.

Another interesting consequence of using "donut" elastomer suspension in pivoted tonearms with offset headshell is side preloading. Without antiskating, the horizontal compliance of the cartridge's suspension would be radically different left and right!

There is one more area where elastomers differ a lot from springs. Exposing a spring to a wide spectrum of frequencies in addition to main pulling/compressing force doesn't change its stress/strain behaviour at all.

Elastomers, on the other hand, "stiffen up" when excited at different frequencies. To put it another way, exposing an elastomer to high frequency vibration "freezes" it for low freqency excitation. For rubber and silicones, such dynamic stiffness is 1.2-1.4 of static.

Among other interesting things, this means that*measuring LF resonance with pure sine frequencies recorded on test LP is completely pointless*.

Elastomers, on the other hand, "stiffen up" when excited at different frequencies. To put it another way, exposing an elastomer to high frequency vibration "freezes" it for low freqency excitation. For rubber and silicones, such dynamic stiffness is 1.2-1.4 of static.

Among other interesting things, this means that

"But wait," you say. "Here are the dynamic and static compliance figures from a cartridge specification, and dynamic is 4 times lower than static, not 1.2-1.4!"

Good question. Most likely, both specifications aren't what they say they are.

Many cartridge manufacturers derive static compliance number from a test with a tonearm of known effective mass and sine signals from a test LP. By applying Mr. Carlson's formula, they come up with a compliance number. As we've seen above, this number isn't necessarily meaningful.

Dynamic compliance, when quoted by Japanese manufacturers, means something else entirely. It's a product of a different test, designed to highlight tracking ability of a cartridge and its relationship with downforce. Nothing to do with LF resonance.

Good question. Most likely, both specifications aren't what they say they are.

Many cartridge manufacturers derive static compliance number from a test with a tonearm of known effective mass and sine signals from a test LP. By applying Mr. Carlson's formula, they come up with a compliance number. As we've seen above, this number isn't necessarily meaningful.

Dynamic compliance, when quoted by Japanese manufacturers, means something else entirely. It's a product of a different test, designed to highlight tracking ability of a cartridge and its relationship with downforce. Nothing to do with LF resonance.

From Audio Technica VM740ML specifications

It's especially interesting because AT quotes very high compliance for VM740ML. In fact, the new generation of AT MM cartridges has a lot lower compliance than the previous one.

Let's sum up what we have discovered so far:

1

Elastomers, as used in stereo cartridge suspension, do not follow Hooke's law. Their deformation is not linearly proportional to applied force

2

This means a harmonic oscillator formula, from which the LF resonance formula is derived, does not apply

3

Rather than single spring constant *k* or compliance *c*, elastomers have 3 distinctly different reaction zones depending on applied force. Regarding cantilever suspension, this means 3 distinct compliances depending on deflection.

4

Vertical compliance differs from horizontal because of downforce preload

5

Stiffness (compliance) of elastomer depends on frequency content of the signal being played back

When applied to modern stereo cartridges, Mr Carlson's 1954 formula can at best be treated as a gross oversimplification

There probably is a way to mathematically describe the results of this investigation, but the necessary physics and maths are, I'm afraid, for me are too complicated and time-consuming.

However, it is clear to me that, when applied to modern stereo cartridges, Mr Carlson's 1954 formula can at best be treated as a gross oversimplification. It might still sort of predict the resonant frequency of the system when it is excited by a pure sine LF signal of a test disc — but in real-life playback of spectrally dense material, the real resonance will differ.

Until someone comes up with a proper calculation accounting for actual elastomer properties, it's probably best to simply ignore the formula and select compliance/effective mass combinations empirically.

However, it is clear to me that, when applied to modern stereo cartridges, Mr Carlson's 1954 formula can at best be treated as a gross oversimplification. It might still sort of predict the resonant frequency of the system when it is excited by a pure sine LF signal of a test disc — but in real-life playback of spectrally dense material, the real resonance will differ.

Until someone comes up with a proper calculation accounting for actual elastomer properties, it's probably best to simply ignore the formula and select compliance/effective mass combinations empirically.

This concludes our little study. I hope that by the time next post comes out, at least our reference playback system would be functional. Can't wait to hear how the flexure-based tonearm sounds!

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